For any two $n \times n$ real symmetric and positive definite matrices $B$ and $C$, is it always possible to find a third real symmetric and positive definite matrix $A$ such that $ABA=C$? If not, give a counter example in a $2 \times 2$ case?
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1Try the $2\times 2$ case yourself ! – Dietrich Burde Sep 12 '14 at 13:39